Molecular Geometry & Bonding
what the shape decides
Series

Defect — the series

11 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The state on a defect at site 31 of 60. The amplitude of one eigenvector at each site of a 60-site chain whose site 31 has its energy raised by 1.6β. The two colours are the two signs. A state belonging to the whole chain would reach across it; this one does not.

    A defect is a level in the gap

    Change one site's energy in a chain of a hundred and sixty and a level leaves the band, carrying a state that lives on a handful of atoms. In one dimension it happens for any change however small — the threshold measured on chains of forty, eighty and a hundred and sixty halves with every doubling, so there is no threshold at all.

    part 1 · solids
  2. The state on the end of a chain of 60. The amplitude of one eigenvector at each site of a 60-site chain whose site 1 has its energy raised by 0.6β. The two colours are the two signs. A state belonging to the whole chain would reach across it; this one does not.

    The end is the hardest place to bind

    A site in the middle of a chain traps a state for any energy difference however small. The site at the end demands a whole β before it traps anything — measured at 1.025, 1.013 and 1.006 on chains of forty, eighty and a hundred and sixty, converging on exactly one. The intuition runs the other way and is wrong.

    part 2 · solids
  3. Two impurities at 2 to 14 sites apart. The splitting between the two levels a pair of impurities of strength -2β pulls out of a chain of 61, against how far apart they are, on a logarithmic scale. It falls by a constant factor per site of separation, and that factor is the decay of the isolated bound state computed from its energy alone. The two levels close on the single impurity's level as the pair separates.

    Two defects, and the level between them

    One impurity pulls a level out of a band at exactly −√(h²+4). Two of them pull out a pair, split by an amount that falls by a factor of 0.41421 for every site of separation — which is √2 − 1, predicted from the isolated level's energy alone and measured to six figures.

    part 3 · solids
  4. A 6×5 patch with one site missing. A 6 by 5 patch of a square structure with one site removed, the two colours of the bipartite structure drawn differently. The disc areas show where the level at zero has its amplitude: entirely on one colour.

    A vacancy is not an impurity

    An impurity is a site whose energy has been changed, and everything about the level it produces depends on by how much. A vacancy is a site that is not there, and the levels it leaves sit at exactly zero for a reason that cannot be tuned, weakened or moved — the count of them is a difference between two numbers of atoms, and two vacancies do not split however far apart they are put.

    part 4 · solids
  5. One impurity is a level; many are a band. The impurity levels of a ring of 160 with sites of depth -3, drawn as a bar from the lowest to the highest, against the fraction of sites that are impurities. At the lowest concentration every level is at the same energy and the bar has no height at all. By 30 per cent the levels span 2.44 and have closed to within 0.25 of the host band, which is shaded.

    One defect is a level, many are a band

    A single deepened site in a chain pulls one state out of the band to −√(h² + 4), exactly, and holds it on 1.42 sites. Put in more and the levels spread: at one site in ten they span 1.45 in the same units and have closed to within 0.69 of the host band, and above one site in eight the count of levels stops matching the count of defects, because two defects on neighbouring sites push one of their pair back into the band.

    part 5 · solids
  6. Where the two halves of an alloy come apart. The gap at the centre of a binary alloy's band against the contrast between its two components, for a chain of 2048 sites arranged three ways. The ordered arrangement's gap is exactly twice the contrast and opens at once; the segregated one's is exactly the contrast less the band width and opens at 2; and the random one, which has no closed form, opens a little before the segregated one and stays a little wider. Below the openings the curves sit at one level spacing rather than at zero, which is what a finite chain has instead of a gap.

    Two bands, if the chain is short enough

    Take a chain, raise half its sites and lower the other half, and ask when the band comes apart into two. The ordered arrangement splits at once, the segregated one at a contrast equal to the band width, and the random one splits earlier than either — and then closes again as the chain is made longer, because a long chain contains a long run of like atoms and a long run is a narrow sub-band.

    part 6 · solids
  7. A state in the gap is a particle in a box the alloy happened to make. The participation ratio of the 40 levels nearest the gap centre, over 5 chains of 400 sites, against the length of the run of like sites each one sits on. The line is 2(L+1)/3, the participation ratio of the ground state of an isolated chain of L sites, with nothing fitted. 38 of 40 lie on it to within 3.1 per cent. The ones above it are states shared between two runs close enough to talk, which is a defect band beginning.

    A particle in a box the alloy made

    A random alloy's gap is set by the longest run of like sites. What sits at the edge of that gap turns out to be the simplest state in quantum mechanics: a particle in a box of L sites, occupying 2(L+1)/3 of them, to within three per cent and with nothing fitted — except for the two states in forty that found a second run to share.

    part 7 · solids
  8. Where the coupling overtakes the spacing. Two quantities that both depend on the chain length, for runs of 6. The coupling is the splitting of the closest pair of runs, which rises as the chain grows because a longer chain brings some pair closer together. The spacing is the mean separation of the gap levels in energy, which falls as the chain grows because there are more of them. They cross at 8,675 sites, and a set of levels coupled more strongly than they are spaced is a band.

    The length at which levels become a band

    Two runs of low sites share a state when they are close enough, and the splitting falls exponentially over two sites. A longer chain brings some pair closer while spreading its levels thinner, so the two quantities run against each other and cross — at about a thousand sites for runs of four and eighty thousand for runs of eight. Chains of four hundred sites are far below that, which is why every gap state on them is a box.

    part 8 · solids
  9. The coupling at the closest pair, and at the typical one. For runs of four to eight low sites at half concentration: the splitting between two runs one site apart, which is the largest coupling any chain can produce, and the splitting at the separation two runs typically have. Both on a logarithmic axis spanning 173 decades. A band width taken from the second is not a small number; it is not a number.

    A band that is a hundred and seventy decades of nothing

    The chain length at which a defect's levels become a band can be located by asking when the coupling between the closest pair exceeds the level spacing. A band also has a width, and a width is set by the typical coupling rather than the closest one. For runs of eight at half concentration those two numbers differ by a hundred and seventy-three orders of magnitude.

    part 9 · solids
  10. The count, for five run lengths. The fraction of runs whose nearest neighbour of the same length is coupled more strongly than a threshold, against how many decades below the closest possible coupling that threshold sits. Every curve is a straight line over this range, because the fraction is small and the geometric tail is linear in the separation there. The slope is what the next figure is about: it is the whole content of the distribution, and it is a product of two numbers.

    A count rather than an average

    Two couplings quoted from one distribution sit a hundred and seventy decades apart. Neither is a summary of it. The quantity that decides how much of a spectrum near a box level is resonant pairs is a count of pairs above a threshold — and it has a closed form, which is a density times a reach times the logarithm of ten.

    part 10 · solids
  11. The resonant share is not a straight line; it bends over and saturates. The share of runs of six in a resonant pair against the logarithm of the chain length, from a thousand sites to 10⁵⁰, at five concentrations of low sites. Dashed lines carry the slope from 10³ to 10¹² straight on. At x = 0.5 the share is 16.1 per cent at 10¹² and 57.7 per cent at 10⁵⁰, where the straight line would be at 78.6. At x = 0.75, the concentration with the most runs of six, it passes 76 per cent by 10³⁰.

    The share that was read as a line

    The share of defect runs sitting in a resonant pair was read as a straight line in the logarithm of the chain length, rising by a density times a reach per decade and never saturating. The exact rise is that amount times the share of runs not yet resonant. At the concentration first studied the difference is small over nine decades and large beyond them; swept to the concentration with the most runs, the rise falls to a third, and half of all runs are resonant ten orders of magnitude later than the straight line says.

    part 11 · solids

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